Theoretical investigation of dynamics and concurrence of entangled \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {P}}}{{\mathcal {T}}}$$\end{document}PT and anti-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {P}}}{{\mathcal {T}}}$$\end{document}PT symmetric polarized photons

Non-Hermitian systems with parity-time \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {(PT)}$$\end{document}(PT) symmetry and anti-parity-time \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {(APT)}$$\end{document}(APT) symmetry have exceptional points (EPs) resulting from eigenvector co-coalescence with exceptional properties. In the quantum and classical domains, higher-order EPs for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {P}}}{{\mathcal {T}}}$$\end{document}PT symmetry and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {APT}$$\end{document}APT-symmetry systems have been proposed and realized. Both two-qubits \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {APT}$$\end{document}APT-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {APT}$$\end{document}APT and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {P}}}{{\mathcal {T}}}$$\end{document}PT-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {P}}}{{\mathcal {T}}}$$\end{document}PT symmetric systems have seen an increase in recent years, especially in the dynamics of quantum entanglement. However, to our knowledge, neither theoretical nor experimental investigations have been conducted for the dynamics of two-qubits entanglement in the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {P}}}{{\mathcal {T}}}$$\end{document}PT-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {APT}$$\end{document}APT symmetric system. We investigate the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {P}}}{{\mathcal {T}}}$$\end{document}PT-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {APT}$$\end{document}APT dynamics for the first time. Moreover, we examine the impact of different initial Bell-state conditions on entanglement dynamics in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {P}}}{{\mathcal {T}}}$$\end{document}PT-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {P}}}{{\mathcal {T}}}$$\end{document}PT, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {APT}$$\end{document}APT-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {APT}$$\end{document}APT and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {P}}}{{\mathcal {T}}}$$\end{document}PT-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {APT}$$\end{document}APT symmetric systems. Additionally, we conduct a comparative study of entanglement dynamics in the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {P}}}{{\mathcal {T}}}$$\end{document}PT-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {P}}}{{\mathcal {T}}}$$\end{document}PT symmetrical system, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {APT}$$\end{document}APT-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {APT}$$\end{document}APT symmetrical system, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {P}}}{{\mathcal {T}}}$$\end{document}PT-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {APT}$$\end{document}APT symmetrical systems in order to learn more about non-Hermitian quantum systems and their environments. Entangled qubits evolve in a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {P}}}{{\mathcal {T}}}$$\end{document}PT-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {APT}$$\end{document}APT symmetric unbroken regime, the entanglement oscillates with two different oscillation frequencies, and the entanglement is well preserved for a long period of time for the case when non-Hermitian parts of both qubits are taken quite away from the exceptional points.


Experimental setup and methodology
In the case of a single qubit, a nontrivial Hamiltonian takes the form 61 here, j = 1, 2 expresses qubit one and two, σ x and σ z are Pauli operators, s j > 0 defines energy scale parameter, r j = γ j s j > 0 describes the degree of non-Hermiticity and ψ j depicts a PT -symmetric and APT -symmetric Hamiltonian by taking ψ j = 0 and ψ j = π 2 , respectively. The eigenvalues of generalized Hamiltonian are calculated as for PT -symmetric ( ψ j = 0 ) the eigenvalues are imaginary for r j > 1 ( PT -symmetric broken regime) and for 0 < r < 1 ( PT -symmetric unbroken regime) eigenvalues are real. On the other hand, for APT -symmetric ( ψ j = π 2 ) the eigenvalues are imaginary for 0 < r j < 1 ( APT -symmetric broken regime) and real for the case r j > 1 ( APT -symmetric unbroken regime). For both scenarios, eigenvalues get zero at exceptional point (EP) (1) H j = e iψ j s j σ x + iγ j σ z = s j e iψ j ir j 1 1 − ir j , (2) E j = ±s j e iψ j 1 − r 2 j , Figure 1. Experimental mechanism. Green area: A 404 nm laser light is passed through a type-I spontaneous parametric down-conversion using a nonlinear barium-borate crystal to generate pairs of 808 nm single photons. Red area: A signal photon exists in a linear polarization state after passing through a 3 nm interference filter (IF). Brick area: Beam displacement devices (BDs) are used in conjunction with half-wave and quarterwave plates (HWPs and QWPs) in order to construct Û j,PT and Û j,APT . Quantum-state tomography is used in the final measurement part to construct the density matrix. Here PBS stands for a polarization beam splitter. Simulating Û j,PT is accomplished by choosing the plate combinations in the dotted black wireframe while simulating Û j,APT by selecting the plates in the solid black wireframe, for more detail please follow 45 . www.nature.com/scientificreports/ r j = 1 . As s j is an energy scale, different quantum states evolve over time under the Hamiltonian Ĥ j,PT (Ĥ j,APT ) at the same rate. In order to ensure generality, we assume that s j = 1 for both Ĥ j,PT and Ĥ j,APT . We define non-unitary operators Û j,PT = exp(−itĤ j,PT ) and Û j,APT = exp(−itĤ j,APT ) , which can be realized in experimental setup as shown in Fig. 1, here we set = 1 . The APT -symmetric non-unitary operator is described as and APT -symmetric non-unitary operator is expressed as where the loss-dependent operator can be depicted as Loss-dependent operators can be realized by combining beam displacers (BDs) and two half-wave plates (HWPs) set at angles η i and η j , for more detail please see 45 . In the above equations, R HWP is defined as the rotation operator of half-wave-plates (HWP) and R QWP is defined as a rotation operator of a quarter-wave plate (QWP), Using an PT -symmetric or APT -symmetric system, we can calculate the total non-Hermitian Hamiltonian of two qubits (1, 2) as Non-unitary operator U(t) = exp(−iĤt) for the two-qubit Hamiltonian can be expressed as follows By using the time-dependent density matrix, we can capture the nonunitary dynamics of combined systems The two qubits in our proposed experiment are two photons with orthogonally polarized states |H� and |V � . As shown in Fig. 1, the initial entangled Bell states of two photons are generated by a spontaneous parametric down-conversion process (left panel), then each photon undergoes an independent time evolution. In this work, we considered two different kinds of initial Bell states: |B 00 � = 1 √ 2 (|00� + |11�) and |B 01 � = 1 √ 2 (|01� + |10�) . We can generate a relative phase between two photons by sandwich structure device (QWP-HWP-QWP) as presented in Fig. 1. In the experiment, the density matrix can be constructed at any time t by quantum state tomography 62 as they pass through the time evolution section. To quantify entanglement between two photons, we calculate the concurrence 63 here, i (i = 1, 2, 3, 4) defines the eigenvalues of the evolution matrix R = ρ σ y ⊗ σ y ρ * σ y ⊗ σ y in decreasing order, where σ y describes the y-Pauli matrix.

Identical experimental construction for both qubits
We dived this section into two parts, wherein the first part we would like to discuss the dynamics of the qubits in an identical PT -PT experimental setup, and in the second part, we emphasize on the finding of the APT -APT experimental setup.
Dynamics of qubits in PT -PT systems. As a first step, we consider the case as illustrated in Eq. (9a) in which both qubits evolve in PT -PT symmetric systems. For this, we explore the time-evolution of entanglement when the two qubits are evolving in a PT -symmetric unbroken regime (r 1 = r 2 = r < 1) . Fig. 2a,c shows the evolution of entanglement when: r = 0.1 (blue curve), (ii) r = 0.5 (black curve), and (iii) r = 0.9 (red curve). Here for the case Fig. 2a we take the initial Bell state |B 00 � = 1 √ 2 (|00� + |11�) and Fig. 2c for the initial Bell state . (|01� + |10�) . In the case of Figs. 2a,c, we note that the concurrence has periodic oscillation with the period T 00 respectively, here for simplicity we take r 1 = r 2 = r . There is a periodic oscillation of the concurrence for initial condition |B 00 �(|B 01 �) during time evolution, with minimal values 0.98 (0.9), 0.60 (0.22) and 0.10 (0.006) for r = 0.1 , r = 0.5 , and r = 0.9 , respectively, at different times, while the peak value for all cases equals 1. We find that the frequency of oscillations increases as parameter r decreases for both cases Figs. 2a,c. To some extent, both cases are identical but the concurrence decreases less for the initial Bell state |B 00 � as compared to |B 01 � and both Bell states have different time periods.
The above results can be illustrated by some physical explanations. We know that r describes the degree of non-Hermiticity of the system, so by decreasing the value of r we increase the non-Hermitian part iσ j of the Hamiltonian. This results in a shorter oscillation period since the energy of the PT -symmetric system increases accordingly. Therefore, we have preservation of entanglement for a longer period of time for small values of r. This entanglement preservation gets strong when we consider the Bell state |B 00 � as an initial condition. Furthermore, we also investigate the dynamics of the entangled photons in the PT -symmetric broken regime (r 1 = r 2 = r > 1) for different initial Bell states |B 00 �(|B 01 �) as presented in Fig. 3a. We note that the sudden death of the entanglement for r 1 = r 2 = 1.1 , and as we increase the value of the non-Hermitian part r the sudden death time decreases. We also report that the entanglement decays slower for the initial Bell state |B 01 � as compared to the |B 00 � as shown in Fig. 3a. At the end of this subsection, we would like to discuss a more general case, when both qubits evolve in different forming, i.e., r 1 = r 2 . Despite non-periodic oscillations in the concurrence, these results are not very different from previous cases, so we have not presented them here to avoid cumbersome paper length, the non-periodic oscillations appear when one qubit is in the broken-symmetry regime and the other one is in the unbroken-symmetry regime. in which initially entangled qubits |B 00 �(|B 01 �) evolve in identically experimental construction APT -APT . In an APT -symmetric unbroken regime, we study the time-evolution of entanglement (r 1 = r 2 = r > 1) , we also used r 1 = r 2 = r here to simplify the analysis. As shown in Fig. 2b,d, entanglement evolution for r = 3 (blue curve), (ii) r = 2 (black curve) and (iii) r = 1.1 (red curve), for the Fig. 2b, we begin with the initial Bell state |B 00 � = 1 √ 2 (|00� + |11�) and for the case Fig. 2d we take the initial Bell state |B 01 � = 1 √ 2 (|01� + |10�) . Based on the following scenario, we note that the concurrence has periodic oscillation with the period T 00 APT = π/(2 √ r 2 − 1) and T 01 APT = π/( √ r 2 − 1) for the Fig. 2b,d respectively. In this special scenario, we also note periodic oscillation of the entanglement for both initial conditions |B 00 �(|B 01 �) for the identically experimental construction APT -APT . We notice that in both cases the frequency of the entanglement oscillations increases as the parameter r increases, however, at the same time, the entanglement amplitude decreases less with increasing r. As we already discussed in the previous part, both cases are identical however we note that the concurrence decreases less for the initial Bell state |B 00 � as compared to |B 01 � . Thus, large values of r lead to the preservation of entanglement for a longer period of time for the initial Bell state |B 00 � . Furthermore, we examine the dynamics of entangled photons in the APT -symmetric broken regime (r < 1) for two unlike initial Bell states |B 00 �(|B 01 �) as shown in Fig. 3b. We observe, the entanglement exponentially ended for r = 0.9 , and as we decrease the value r the sudden death time decreases. Similar to the case PT -PT , here also our findings show that the entanglement decays slower for the initial Bell state |B 00 � versus |B 01 � as predicted in Fig. 3b.

Different PT -APT experimental construction for both qubits
In this section, we discuss in detail, unlike experimental construction for both qubits, which is in contradistinction to the previous section. Here, we consider that the first qubit passes through the PT experimental setup, and the second qubit moves through the APT experimental setup. In this regard, Fig. 4 presents the dynamical evolution of the entanglement. For the first case, we consider r 1 = 0.9 and r 2 = 1.1 near the exceptional point. We note that the sudden decay of entanglement and later revives is shown in Fig. 4a. In another scenario, the first qubit, we move away from the exceptional point r 1 = 0.1 , and the second qubit is still near the exceptional point r 2 = 1.1 as shown in Fig. 4a with a black square line. We notice that the entanglement decays however do not have sudden death and complete revivals can also be seen as plotted in Fig. 4a with a black-square line. As both qubits move away from the exceptional points, we find that the entanglement does not decay but starts to oscillate with some frequencies as shown in Fig. 4a,b,c. We note that the entanglement is well preserved if we move away from the exceptional points in the unbroken regimes of the PT -symmetric system as given in Fig. 4b and for the APT -symmetric system as shown in Fig. 4c. With this, we conclude that entanglement can be well preserved when we consider our experimental parameters quite away from the exceptional points. Theoretically, for this special case, it is hard to investigate the oscillation frequency of the entanglement as the dynamics of the entanglement get complex. Here, we consider only one initial Bell state |B 01 � as we do not find much difference in the entanglement for the second initial condition |B 00 � . Therefore, to avoid repetition in results, we do not present them here. However, we note a delay in the sudden death of the entanglement for the initial Bell state |B 00 � as compared to the Bell state |B 01 � as predicted in Fig. 3c. We also calculate the dynamics of entanglement for the PT and APT -symmetric systems in broken and unbroken regimes, respectively as shown in Fig. 5. We notice non-periodic oscillations of entanglement as predicted in Fig. 5a,b. We note that the broken PT -symmetric regime has a strong effect as compared to the unbroken APT -symmetric regime, therefore the entanglement  www.nature.com/scientificreports/ decays as predicted in Fig. 5. From Fig. 5, we also study that the initial Bell state |B 00 � decays fast as compared to the initial Bell state |B 01 �.

Summary and conclusion
Quantum states in the PT -symmetric system, as well as the APT -symmetric system, are evolvable by applying the non-unitary evolution operator, we examine how entangled states evolve over time. The nonunitary operator is implemented by decomposing it into unitary matrices and loss-dependent operators. We can achieve the PT -symmetric system as well as the APT -symmetric system by linear optical elements. All nonunitary operators in the PT -symmetric systems and APT -symmetric systems can be realized using this approach. This report explores how entanglement between two qubits evolves over time in PT -PT symmetric system, APT -APT symmetric system, and PT -APT symmetric system. Non-periodic oscillations, Periodic oscillations, delayed vanishing, rapid decay, and sudden death of entanglement are all observed in our theoretical simulations for different initial conditions and for various system parameters. We noted that the entanglement oscillates periodically when the non-Hermitian part r 1 = r 2 = r is taken quite away from the exceptional point for both cases, i.e., PT -PT symmetric and APT -APT symmetric unbroken regimes. We also observed that in both  www.nature.com/scientificreports/ cases ( PT -PT and APT -APT ) the frequency of the entanglement oscillations increase as the parameter r is taken away from exceptional points, and at the same time, the entanglement amplitude decreases less. This shows that the entanglement is well-preserved for these scenarios. Nonperiodic oscillations of Entanglement were detected when both qubits evolve through in different forming, i.e., r 1 = r 2 for both cases, i.e., PT -PT symmetric and APT -APT symmetric unbroken regimes. We also noted that as we moved near the exceptional point, the entanglement decayed rapidly and later revived. However, for the PT -PT symmetric and APT -APT symmetric broken regimes, we found the entanglement of sudden death. It is really an interesting finding that the entanglement survives for a longer period of time for the Bell state |B 00 � as compared to the Bell state |B 01 � . We also observed that the entanglement time periods T 00 PT = π/(2 √ 1 − r 2 ) ( T 00 APT = π/(2 √ r 2 − 1) ) and T 01 PT = π/( √ 1 − r 2 )(T 01 APT = π/( √ r 2 − 1) ) are different for different Bell initial conditions |B 00 � and |B 01 � , respectively, for PT (APT )-symmetric unbroken regimes. When both qubits evolve different experimental construction i.e., PT -APT symmetric unbroken regime, we noted that the entanglement oscillates with two different oscillation frequencies and the entanglement is well protected for the case when both qubits' non-Hermitian parts are taken quite away from the exceptional points. For this special case, the dynamics of the entanglement get complex therefore it is not possible to find a theoretical formula for the time period of entanglement oscillations. By examining the phenomena found in this scientific report, we can gain a better understanding of quantum open systems. In addition to PT -PT symmetric systems, APT -APT symmetric systems, PT -APT symmetric systems, and other non-Hermitian quantum systems, the present work opens up a new area for future studies in quantum entanglement dynamics in multiqubit systems.

Data availibility
The datasets generated during and/or analyzed during the current study are available from the corresponding author upon reasonable request.